Management of portfolio depletion risk through optimal life cycle asset allocation. (English) Zbl 1426.91218

Summary: Members of defined contribution (DC) pension plans must take on additional responsibilities for their investments, compared to participants in defined benefit (DB) pension plans. The transition from DB to DC plans means that more employees are faced with these responsibilities. We explore the extent to which DC plan members can follow financial strategies that have a high chance of resulting in a retirement scenario that is fairly close to that provided by DB plans. Retirees in DC plans typically must fund spending from accumulated savings. This leads to the risk of depleting these savings, that is, portfolio depletion risk. We analyze the management of this risk through life cycle optimal dynamic asset allocation, including the accumulation and decumulation phases. We pose the asset allocation strategy as an optimal stochastic control problem. Several objective functions are tested and compared. We focus on the risk of portfolio depletion at the terminal date, using such measures as conditional value at risk (CVAR) and probability of ruin. A secondary consideration is the median terminal portfolio value. The control problem is solved using a Hamilton-Jacobi-Bellman formulation, based on a parametric model of the financial market. Monte Carlo simulations that use the optimal controls are presented to evaluate the performance metrics. These simulations are based on both the parametric model and bootstrap resampling of 91 years of historical data. The resampling tests suggest that target-based approaches that seek to establish a safety margin of wealth at the end of the decumulation period appear to be superior to strategies that directly attempt to minimize risk measures such as the probability of portfolio depletion or CVAR. The target-based approaches result in a reasonably close approximation to the retirement spending available in a DB plan. There is a small risk of depleting the retiree’s funds, but there is also a good chance of accumulating a buffer that can be used to manage unplanned longevity risk or left as a bequest.


91G05 Actuarial mathematics
91G10 Portfolio theory
93E20 Optimal stochastic control
Full Text: DOI


[1] Arnott, R. D.; Sherrerd, K. F.; Wu., L., The glidepath illusion and potential solutions, Journal of Retirement, 1, 2, 13-28 (2013)
[2] Basak, S.; Chabakauri., G., Dynamic mean-variance asset allocation, Review of Financial Studies, 23, 2970-3016 (2010)
[3] Björk, T.; Murgoci, A.; Zhou., X. Y., Mean variance portfolio optimization with state dependent risk aversion, Mathematical Finance, 24, 1-24 (2014) · Zbl 1285.91116
[4] Blake, D.; Cairns, A. J. G.; Dowd., K., Pensionmetrics 2: Stochastic pension plan design during the distribution phase, Insurance: Mathematics and Economics, 33, 29-47 (2003) · Zbl 1043.62086
[5] Blake, D.; Wright, D.; Zhang., Y., Age-dependent investing: Optimal funding and investment strategies in defined contribution pension plans when members are rational life cycle financial planners, Journal of Economic Dynamics and Control, 38, 105-124 (2014) · Zbl 1402.90200
[6] Cairns, A. J. G.; Blake, D.; Dowd., K., Stochastic lifestyling: Optimal dynamic asset allocation for defined contribution pension plans, Journal of Economic Dynamics and Control, 30, 843-977 (2006) · Zbl 1200.91297
[7] Campanele, C.; Fugazza, C.; Gomes., F., Life-cycle portfolio choice with liquid and illiquid financial assets, Journal of Monetary Economics, 71, 67-83 (2015)
[8] Chen, A.; Delong., L., Optimal investment for a defined-contribution pension scheme under a regime switching model, ASTIN Bulletin, 45, 397-419 (2015) · Zbl 1390.91168
[9] Christiansen, M. C.; Steffensen., M., Around the life cycle: Deterministic consumption-investment strategies, North American Actuarial Journal (2018) · Zbl 1416.91345
[10] Cocco, J. F.; Gomes, F. J.; Maenhout., P. J., Consumption and portfolio choice over the life cycle, Review of Financial Studies, 18, 491-533 (2005)
[11] Cogneau, P.; Zakalmouline., V., Block bootstrap methods and the choice of stocks for the long run, Quantitative Finance, 13, 1443-1457 (2013) · Zbl 1281.91141
[12] Cong, F.; Oosterlee., C. W., Multi-period mean variance portfolio optimization based on Monte-Carlo simulation, Journal of Economic Dynamics and Control, 64, 23-38 (2016) · Zbl 1401.91513
[13] Cont, R.; Mancini., C., Nonparametric tests for pathwise properties of semimartingales, Bernoulli, 17, 781-813 (2011) · Zbl 1345.62074
[14] Dahlquist, M.; Setty, O.; Vestman., R., On the asset allocation of a default pension fund, Journal of Finance, 73, 1893-1936 (2018)
[15] Dammon, R. M.; Spatt, C. S.; Zhang., H. H., Optimal asset location and allocation with taxable and tax-deferred investing, Journal of Finance, 59, 999-1037 (2004)
[16] Dang, D.-M.; Forsyth., P. A., Continuous time mean-variance optimal portfolio allocation under jump diffusion: A numerical impulse control approach, Numerical Methods for Partial Differential Equations, 30, 664-698 (2014) · Zbl 1284.91569
[17] Dang, D.-M.; Forsyth., P. A., Better than pre-commitment mean-variance portfolio allocation strategies: A semi-self-financing Hamilton-Jacobi-Bellman equation approach, European Journal of Operational Research, 250, 827-841 (2016) · Zbl 1348.91250
[18] Dang, D.-M.; Forsyth, P. A.; Vetzal., K. R., The 4 · Zbl 1402.91682
[19] Dichtl, H.; Drobetz, W.; Wambach., M., Testing rebalancing strategies for stock-bond portfolos across different asset allocations, Applied Economics, 48, 772-788 (2016)
[20] Donnelly, C.; Gerrard, R.; Guillén, M.; Nielsen, J. P., Less is more: Increasing retirement gains by using an upside terminal wealth constraint, Insurance: Mathematics and Economics, 64, 259-267 (2015) · Zbl 1348.91251
[21] Donnelly, C.; Guillén, M.; Nielsen, J. P.; Pérez-Marín., A. M., Implementing individual savings decisions for retirement with bounds on wealth, ASTIN Bulletin, 48, 111-137 (2017) · Zbl 1390.91178
[22] Fagereng, A.; Gottlieb, C.; Guiso., L., Asset market participation and portfolio choice over the life-cycle, Journal of Finance, 72, 705-750 (2017)
[23] Forsyth, P.; Labahn., G., \(####\)-monotone Fourier methods for optimal stochastic control in finance, Journal of Computational Finance, (2019)
[24] Forsyth, P. A.; Vetzal., K. R., Dynamic mean variance asset allocation: Tests for robustness, International Journal of Financial Engineering, 4, 2, 1750021 (2017)
[25] Forsyth, P. A.; Vetzal., K. R., Robust asset allocation for long-term target-based investing, International Journal of Theoretical and Applied Finance, 20, 3, 1750017 (2017) · Zbl 1396.91686
[26] Forsyth, P. A.; Vetzal, K. R., Optimal asset allocation for retirement savings: Deterministic vs. time conistent adaptive strategies (2018)
[27] Forsyth, P. A.; Vetzal, K. R.; Westmacott, G., Target wealth: The evolution of target date funds. White paper, PWL Capital (2017)
[28] Freedman, B., Efficient post-retirement asset allocation, North American Actuarial Journal, 12, 228-241 (2008)
[29] Gao, J.; Zhou, K.; Li, D.; Cao., X., Dynamic mean-LPM and mean-CVaR portfolio optimization in continuous-time, SIAM Journal on Control and Optimization, 55, 1377-1397 (2017) · Zbl 1414.91338
[30] Gerrard, R.; Haberman, S.; Vigna., E., Optimal investment choices post-retirement in a defined contribution pension scheme, Insurance: Mathematics and Economics, 35, 321-342 (2004) · Zbl 1093.91027
[31] Gerrard, R.; Haberman, S.; Vigna., E., The management of decumulation risk in a defined contribution pension plan, North American Actuarial Journal, 10, 84-110 (2006)
[32] Graf, S., Life-cycle funds: Much ado about nothing?, European Journal of Finance, 23, 974-998 (2017)
[33] Guan, G.; Liang., Z., Optimal management of DC pension plan in a stochastic interest rate and stochastic volatility framework, Insurance: Mathematics and Economics, 57, 58-66 (2014) · Zbl 1304.91193
[34] Horneff, V.; Maurer, R.; Mitchell, O. S.; Rogalla., R., Optimal life cycle portfolio choice with variable annuities offering liquidity and investment downside protection, Insurance: Mathematics and Economics, 63, 91-107 (2015) · Zbl 1348.91147
[35] Kou, S. G.; Wang., H., Option pricing under a double exponential jump diffusion model, Management Science, 50, 1178-1192 (2004)
[36] Li, D.; Ng., W.-L., Optimal dynamic portfolio selection: Multiperiod mean-variance formulation, Mathematical Finance, 10, 387-406 (2000) · Zbl 0997.91027
[37] Liang, X.; Young., V. R., Annuitization and asset allocation under exponential utility, Insurance: Mathematics and Economics, 79, 167-183 (2018) · Zbl 1401.91166
[38] Ma, K.; Forsyth, P. A., Numerical solution of the Hamilton-Jacobi-Bellman formulation for continuous time mean variance asset allocation under stochastic volatility, Journal of Computational Finance, 20, 1, 1-37 (2016)
[39] Macdonald, B.-J.; Jones, B.; Morrison, R. J.; Brown, R. L.; Hardy., M., Research and reality: A literature review on drawing down retirement financial savings, North American Actuarial Journal, 17, 181-215 (2013) · Zbl 1412.91050
[40] Mancini, C., Non-parametric threshold estimation models with stochastic diffusion coefficient and jumps, Scandinavian Journal of Statistics, 36, 270-296 (2009) · Zbl 1198.62079
[41] Menoncin, F.; Vigna., E., Mean-variance target based optimisation for defined contribution pension schemes in a stochastic framework, Insurance: Mathematics and Economics, 76, 172-184 (2017) · Zbl 1396.91307
[42] Michaelides, A.; Zhang., Y., Stock market mean reversion and portfolio choice over the life cycle, Journal of Financial and Quantitative Analysis, 52, 1183-1209 (2017)
[43] Milevsky, M. A.; Young., V. R., Annuitization and asset allocation, Journal of Economic Dynamics and Control, 31, 3138-3177 (2007) · Zbl 1163.91440
[44] Miller, C.; Yang., I., Optimal control of conditional value-at-risk in continuous time, SIAM Journal on Control and Optimization, 55, 856-884 (2017) · Zbl 1366.49031
[45] O’Hara, M.; Daverman, T., Reexamining “to versus through.” White Paper, BlackRock (2017)
[46] Patton, A.; Politis, D.; White., H., Correction to: Automatic block-length selection for the dependent bootstrap, Econometric Reviews, 28, 372-375 (2009) · Zbl 1400.62193
[47] Peijnenburg, K.; Nijman, T.; Werker., B. J. M., The annuity puzzle remains a puzzle, Journal of Economic Dynamics and Control, 70, 18-35 (2016) · Zbl 1401.91181
[48] Politis, D.; White., H., Automatic block-length selection for the dependent bootstrap, Econometric Reviews, 23, 53-70 (2004) · Zbl 1082.62076
[49] Ritholz, B., Tackling the ‘nastiest, hardest problem in finance.’ (2017)
[50] Rockafellar, R. T.; Uryasev., S., Optimization of conditional value-at-risk, Journal of Risk, 2, 21-42 (2000)
[51] Rupert, P.; Zanella., G., Revisiting wage, earnings, and hours profiles, Journal of Monetary Economics, 72, 114-130 (2015)
[52] Smith, G.; Gould., D. P., Measuring and controlling shortfall risk in retirement, Journal of Investing, 16, 82-95 (2007)
[53] Strub, M., Li, D., Cui, X., and Gao., J.2017. Discrete-time Mean-CVaR portfolio selection and time-consistency induced term structure of the CVaR. SSRN abstract 3040517. · Zbl 1425.91395
[54] Tretiakova, I.; Yamada, M. S., What DC plan members really want, Rotman International Journal of Pension Management, 4, 60-70 (2011)
[55] Vigna, E., On efficiency of mean-variance based portfolio selection in defined contribution pension schemes, Quantitative Finance, 14, 237-258 (2014) · Zbl 1294.91168
[56] Vigna, E., Tail optimality and preferences consistency for intertemporal optimization problems (2017)
[57] Waring, M.; Siegel., L., The only spending rule you will ever need, Financial Analysts Journal, 71, 91-107 (2015)
[58] Westmacott, G.; Daley, S., The design and depletion of retirement portfolios. White paper, PWL Capital (2015)
[59] Wu, H.; Zeng., Y., Equilibrium investment strategy for defined-contribution pension schemes with generalized mean-variance criterion and mortality risk, Insurance: Mathematics and Economics, 64, 396-408 (2015) · Zbl 1348.91262
[60] Yao, H.; Lai, Y.; Ma, Q.; Jian., M., Asset allocation for a DC pension fund with stochastic income and mortality risk: A multi-period mean-variance framework, Insurance: Mathematics and Economics, 54, 84-92 (2014) · Zbl 1291.91200
[61] Zhang, R.; Langrené, N.; Tian, Y.; Zhu, Z.; Klebaner, F.; Hamza, K., Sharp target range strategy with dynamic portfolio selection: Decensored least squares Monte Carlo (2017)
[62] Zhou, X. Y.; Li., D., Continuous-time mean-variance portfolio selection: A stochastic LQ framework, Applied Mathematics and Optimization, 42, 19-33 (2000) · Zbl 0998.91023
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